AP Physics 1 Fluids Dynamics Speed Review

Buoyancy: Why Some Things Float and Others Sink

Ever tried to push a beach ball underwater? The harder you push, the stronger the water pushes back. That’s buoyancy in action.

Archimedes figured out that the buoyant force equals the weight of the displaced fluid (\(\text{DF}\)):

\[ F_b = m_{DF}g = \rho_{DF} V_{DF} g \]

So the further you push down the beach ball, the more water it will displace, creating a larger buoyant force.

Note: buoyant force does NOT come from the weight of the OBJECT itself.

If \(F_b\) is greater than the object’s weight, it rises until \(F_b\) equals its weight, then floats. If \(F_b\) is less than its weight even when it is fully underwater, down it goes.

This also explains why ships, despite being made of heavy metal, don’t sink: they displace enough water to generate a buoyant force equal to their own weight.

Problem Solving With Buoyant Forces

Follow the forces problem framework:

1. Draw an FBD. Keep in mind fluids, including air, have weight!
2. Sum up forces and use newton’s law \(F = ma\)
3. Derive! If you need, plug in the formula for buoyancy and solve for the unknown.

Try it out this question:

Fluid Flow: Conservation Of Mass

Fluids follow a strict rule: what goes in must come out. That’s where the continuity equation comes in:

\[ A_1 v_1 = A_2 v_2 \]

Here’s what it really means: the volume of fluid flowing per second stays the same, no matter how the pipe’s shape changes. This is called the flow rate (\( Q \)), defined as:

\[ Q = A v \]

Where:

• \(Q\) = flow rate \(\frac{m^3}{s}\)
• \(A\) = cross-sectional area \(m^2\)
• \(v\) = velocity of the fluid \(\frac{m}{s}\)

If a pipe gets narrower, the velocity increases to maintain the same flow rate.

If it widens, the velocity decreases.

That’s why river rapids form when a wide river suddenly narrows—water speeds up to keep the same volume moving through.

The flow rate always stays constant for an incompressible fluid. If area changes, velocity adjusts to compensate.

Try it out:

Bernoulli’s Equation: Conservation of Energy (Revisited)

Bernoulli’s principle is just conservation of energy applied to fluids. It states that a fluid’s total energy (pressure, gravitational potential, and kinetic) stays constant:

\[ P_1 + \rho g y_1 + \frac{1}{2} \rho v_1^2 = P_2 + \rho g y_2 + \frac{1}{2} \rho v_2^2 \]

In short, a gain in one form of energy must come from a decrease in another.

For example, consider what happens when a pipe narrows: the fluid speeds up (according to continuity). But that increase in kinetic energy has to come from somewhere — and it comes from pressure.

This explains:

• How airplanes generate lift (faster air over top of wing = lower pressure at top compared to bottom)
• Why a shower curtain gets sucked inward when you turn on the water
• How a perfume atomizer sprays liquid

Try it out:

Torricelli’s Theorem: Fluid Speed Without the Drama

When a fluid drains out of a hole, it acts like it’s in free fall. Torricelli’s equation gives its speed:

\[ v = \sqrt{2g\Delta y} \]

The greater the height difference, the faster the fluid moves. That’s why water from a tall tower shoots out faster than from a short one.

Final Takeaways (a.k.a. Your Cheat Sheet)

• Fluid pressure increases with depth: \( P = \rho gh \)
• Buoyant force = weight of displaced fluid: \( F_b = \rho V g \)
• Fluid flow speeds up in narrow spaces: \( A_1 v_1 = A_2 v_2 \)
• Fast-moving fluids have low pressure: Bernoulli’s equation
• Fluid draining from a hole follows \( v = \sqrt{2g\Delta y} \)

Master these, and you’ll crush any fluids question the AP throws at you.

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